Tag Archives: Puzzle

Another interesting museum located in the Collegius Maius of the Jagiellonian University is an exhibition about mathematics where children can play and learn a lot! There are old calculators from the 20th century…

Photography by Carlos Dorce

…abacus and slide rulers:

Photography by Carlos Dorce

Children can play with Geography and learn that straight lines in a map are not the shortest ways for the planes:

Photography by Carlos Dorce

They can also learn the theorem of Pythagoras scrolling this interesting figure:

Photography by Carlos Dorce

There are polyhedra and a lot of geometrical and topological games:

Photography by Carlos Dorce

The museum is very small but all the tourist are inside Collegius Maius so you can be very quiet watching all the exhibited objects and toys, like the Rodin’s Thinker:

Photography by Cristina Martínez

Finally… here you have my two children playing with Eulerian graphs! They are lovely! Aren’t they?

Last Wednesday I went to MMACA (Museum of Mathematics of Catalonia) with some of my students. This museum is located in Mercader Palace in Cornellà de Llobregat (near Barcelona) since February and we enjoyed a very interesting “mathematical experience”.

Mercader PalacePhotography by Carlos Dorce

The museum is not so big but you can “touch” and discover Mathematics in all its rooms. I think that there are enough experiences to enjoy arithemtical and geometrical properties, simmetries, mirrors, impossible tessellations, Stadistics,…

Photography by Carlos Dorce

For example, students could check the validity of theorem of Pythagoras in two ways. First of all, they coud weigh wooden squares and check that the square constructed on the hypotenuse of a right triangle weighs the same as the two squares constructed on the other two sides of the triabgle. Later, they discovered that the first square could be divided in some pieces of Tangram with which they could construct the other two squares. So the visitors demonstrated the theorem in a very didactic way: playing with balances and playing with tangram.

Photography by Carlos Dorce

Students also learnt some properties of the cycloid and they could check its brachistochronic characteristic. I imagine Galileo or some of Bernoulli brothers in the 17th century doing the same experiments with a similar instrument. What a wonderful curve! The ball always reaches the central point in the same time and its initial position doesn’t matter!

Photography by Carlos Dorce

Another of the studied curves is the catenary which is one of the emblematic mathematical symbols of Antoni Gaudi’s architecture in Barcelona.

Photography by Carlos Dorce

Of course, polyhedra are very important in the exhibition and visitors can play with them so they discover some of their most important properties. For example, which is the dual polyhedron of the dodecahedron? Playing with it the students could see that the hidden polyhedron is a… You must visit MMACA and discover it!

Photography by Carlos Dorce

Another example: look at these three wooden pieces…

The dodecahedron has an ortonormal symmetry and we can check it with an ortonormal set of mirrors:

Photography by Carlos Dorce

There are more mirrors and more wooden pieces to play and construct other different Platonic and Archimedian polyhedra.

And… did you know that it’s possible to draw a right line playing with two circles? If the red circle rotates within the black one… what figure is described by the yellow point?

Photography by Carlos Dorce

In the 13th century, the great Nasîr al-Dîn al-Tûsî had to build one similar instrument to improve the astronomical geometrical systems with his “Al-Tûsî’s pair”:

Source: Wikimedia Commons

Rotating a circle within another one, he could move a point in a right line without denying Aristotelian philosophy. This dual system was used by al-Tûsî in his Zîj-i Ilkhanî (finished in 1272) and Nicolas Copernicus probably read this innovation together with other Arabic astronomical models. Thinking about them, he began to improve the astronomical system of his De Revolutionibus (1543). Al-Tûsî’s pair was very famous until the 15th century.

In Erathostenes Room there are some Sam lloyd’s puzzles, games about tesselations, Stadistics, Probablility and this quadric:

Photography by Carlos Dorce

I didn’t know that it could be described only with a multiplication table! Is its equation z = xy? Yes, of course! My students also played to build the famous Leonardo’s bridge and they could see that there isn’t necessary any nail to hold a bridge.

Photography by Carlos Dorce

Ah! And I can’t forget to say that if you visit MMACA with a person that don’t like Maths, he/she can always admire this beautiful XIX century Mercader Palace:

Photography by Carlos Dorce

Furthermore, one of the rooms of the palace is decoratd by a chess lover!

The Estonian History Museum is one of the most interesting attractions of Tallinn. We can read in the official web site that the story of the Museum began in 1802, when Tallinn’s town hall pharmacist, Johann Burchard(1776–1838), started a collection called Mon Faible (“My weakness”). The first exhibit was a Chinese opium pipe and Burchard put on the exhibition “Antiquities and rarities” at the House of the Brotherhood of the Blackheads in 1822. Twenty years later, the Estonian Literature Society was founded in Tallinn and one of its aims was to establish a museum “to broaden our knowledge of this country by studying its history, art, manufacturing, technology and nature”. Extensive collections were compiled over the following twenty years, which formed the basis of the Provincial Museum of the Estonian Literature Society, founded in 1864 at the house of St. Canute’s Guild.

In 1911, the Estonian Literature Society purchased premises on Toompea at 6 Kohtu Street, where the museum’s innovative activities could flourish. As the only museum in the city, it became an important focal point in Tallinn’s cultural life with educational lectures and exhibitions. The Museum retained its important position through its valuable collections although the Estonian National Museum in Tartu (founded in 1909) became the most important museum in the Republic of Estonia. In 1940 Estonia was incorporated to the U.S.S.R. and The museum was nationalized and the History Museum of the Estonian Soviet Socialist Republic was established in its place. Overbearing ideological pressure ruined the museum in the following years. In addition to subjugating the museum employees, items that were deemed harmful were eliminated, which meant destroying everything that reminded people of the republic of Estonia. The Museum preserved the greater part of the main collection and it moved to its current location at the Great Guild Hall in 1952. Finally, in 1989 the Museum was renamed the Estonian History Museum and many important exhibitions that introduced the contemporary history of Estonia were held in the late 1980s and early 1990s.

Estonia is a very young country and in the main exhibition we find a lot of references to its former background and to the Soviet occupation. Among all these interesting object there also are some references to mathematical objects. For example, we can see different scholar material as two geometrical figures…

Two geometrical figures (1920-1930)Photography by Carlos Dorce

and a mathematical ruler:

Mathematical ruler (c.1950)Photography by Carlos Dorce

The this section of the exhibition about the schools we can read that the first educated Estonians were said to be monk Nicolaus, who was appointed by the Pope to carry out missionary work in Estonia in 1170, and the parish priest Johannes, who worked with the Livonians at the beginning of the 13th century. In the mid-13th century cathedral schools were established in Tartu, Pärnu, Tallinn and Haapsalu. Next to the monasteries, there were monastic schools, and in the 15th and 16th centuries, town schools were founded. The 17th century was of great importance to Estonian cultural history because it was when that high schools, along with print shops, were first founded in the country. The University of Tartu was founded in 1632. In the 1680s a network of village schools was created but regular school education in rural areas only took hold in the early 19th century. Peasant schools were mandatory, and students were taught in the native language; however, most of the children were allowed to study at home. Writing and arithmetic were taught starting in the mid-19th century. The 1897 census revealed that 94% of Estonians were able to read, which was the highest percentage among the nations of the Russian Empire. The Imperial University of Tartu reopened in 1802 and became a centre of science and intellectuals in the Baltic provinces. In 1803 the university employed an Estonian and a Latvian-language lecturer. On December 1, 1919, the University of Tartu of the Republic of Estonia was opened, marking the beginning of regular higher education in the Estonian language.

Another interesting object is in the lower lever: it’s an abacus which was taken from Ivan Mazepa’s (1639-1709) tent after he Battle of Poltava in 1709. The wooden box has a framed mirror and an abacus with beads made of white and red bone. Mazepa wanted to unite the Ukranian terriories into a single state. During the Great Northern War he initially supported Russia but later changed sides and lost the Battle of Poltava alongside Swedish troops.

Mazepa’s abacusPhotography by Carlos Dorce

Finally, there is a three-dimensional wooden puzzle made in 1920s or 1930s. It consists in 16 pieces. Different wooden puzzles were made in Estonia already in the 19th century:

Estonian Puzzle (c.1925)Photography by Carlos Dorce

As we can see, the Estonian History Museum is very interesting for a Mathematical lover. Furthermore, there is another object which must be mentioned but I’m going to talk about it in the next post.